"Let me be as clear as possible," Obama said. "I think people's families are off-limits, and people's children are especially off-limits. This shouldn't be part of our politics. It has no relevance to Gov. Palin's performance as governor or her potential performance as a vice president."

Obama said reporters should "back off these kinds of stories" and noted that he was born to an 18-year-old mother.

"How a family deals with issues and teenage children, that shouldn't be the topic of our politics, and I hope that anybody who is supporting me understands that's off-limits." Watch more of Obama's comments »

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The aide said the Palins and the McCain campaign decided to reveal the information now because of rampant Internet rumors that Sarah Palin's 4-month-old baby, who has Down syndrome, was actually Bristol's.

Obama became annoyed when asked about a Reuters news service report that quoted an unnamed senior McCain aide saying that Obama's name appears in liberal blogs speculating about Trig's parentage "in a way that certainly juxtaposes themselves against their 'campaign of change.' " Watch analysts weigh in on the Palins' announcement »

"I am offended by that statement," the Illinois senator retorted, not letting the reporter finish his question. "There is no evidence at all that any of this involved us.

"We don't go after people's families; we don't get them involved in the politics. It's not appropriate, and it's not relevant," he added. "Our people were not involved in any way in this, and they will not be. And if I ever thought that there was somebody in my campaign that was involved in something like that, they'd be fired." iReport.com: Share your thoughts on Palin